2001 Hyundai Accent L 1.5 L. Hatchback Power steering pump noise /whine
 

My power steering pump is making this High whining sound . Mechanic said fluid needed to be changed and was going to cost $200 tho $400 to do it
Can we just go it ourselves at home ?
If so. How would that be done ?

Sometimes if the fluid level has run low it will get air into the system and make noise. Bringing the level to the proper height as shown on the dipstick will sometimes allow the air to go out and the noise goes away.
I believe you should get another opinion from a different "mechanic" .

Video on how to DIY.
Watch the whole thing before attempting
And always remember to use the proper power steering fluid for your system
However, I suspect that the system needs to have the power steering pump serviced or replaced
Don't know ... just a guess.